Transcript Isostasy chap 9 LECT..

Isostasy answers (?) these questions
•Is the deep earth rigid as a ‘steel ball’ or is it able to flow viscously like a ‘sea of lava’ ?
•What processes create the low standing oceans and the high standing continents ?
•What forces hold-up the mountains or conversely holds-down basins ?
•What is the earth’s figure best represented as: a sphere or a flattened ellipsoid ?
•What causes the stranded shorelines Scandanavia (uplift evidence) ?
•How does measured gravity relate to the mode of compensation of the earth surface?
•How strong in compression/tension/shear is the earth’s crust and lithosphere ?
•What are the rheologic laws that govern the deformation (stress/strain) of the shallow
and deep portions of the planet ?
Mis-conceptions about crust, lithosphere, asthenosphere*
(1) Crust (compositional). The crust is the
residue from melting the mantle.
Crustal thicknesses is 10 km (oceans)
and 30-80 km (continents).
(2) Mantle (compositional). The mantle
+crust+core = chondrite meteorites.
(3) Core (compositional). Made of
liquid/solid iron mostly.
(4) Lithosphere (strength). Also, called a
plate. The strong layer that slides over
the asthensophere. The lithosphere is
strong because it is colder.
(5) Asthenosphere (strength). The mantle
just below the lithosphere is often
weaker than the deeper mantle due
to pressure and temperature effects.
Pressure gradient and gravity
Early evidence and thoughts on Isostasy
Hypsometry (distribution of elevations) of planet earth
How would gravity vary over small and large mountains ?
10 km
100 km
Remember, a fundamental property of mass is that it has gravity (and inertia). So, the mass
of the two mountains (above) above the surface will increase the gravity recorded by the
gravimeter in the balloon. But, what about the lower density root ? What is important is two
questions: 1) is the crust strong enough to support the weight of a mountain? 2) if the crust
‘breaks’, then what forces support the weight of the mountains? Note that mountain (A)
has no change in density beneath the level, but mountain (C) has a ‘lower density root’
under it.
In general, we’ll find that there are two end-member processes that support or compensate
the weight of the mountains: the strength of the crust for small mountains, and a bouyant
low density root beneath large mountains.
Earth model: an elastic lithosphere (crust), that does not
flow (but does break), over a ‘fluid’ asthenosphere/mantle
Historically, whether the earth viscously flowed has been vigorously debated from 1500
until 1930-1960. Some people said the earth’s interior was as strong as a steel ball. They
were wrong! Other said, the earth’s interior was liquid. They were right (outer core) and
wrong (mantle). The truth is that the mantle is 99.9% solid AND it does flow at 1-10 cm/yr
(10-100 km/Ma) rates. This is called convection. It might take millions of years to move
things around, but the earth is 4,500 million years old!
Local versus regional compensation
The Lithosphere (crust) is strong
enough to support the load (weight)
of the mountain. But, the lithospheric
strength is finite and the surface of
the lithosphere is bowed down
‘regionally’ to support the load.
The Lithosphere (crust) is NOT strong enough to
support the load (weight) of the mountain. In
fact, in the limit, the lithosphere is broken on
either side of the load and has a near zero
strength. The load is supported by the hydrostatic
pressure of the asthenosphere pushing on the
bottom of the loaded block.
Gravity effect
•Positive free air anomaly over load
•Near zero Bouguer anomaly over
load
Gravity effect
•Zero free air anomaly over load
•Negative Bouguer anomaly over load
Regional compensation Plate flexure examples
Regional compensation of Hawaiian Island Chain
Note the strong correlation between the bathymetry
and the free air gravity profile. This is because this
topography is NOT isostatically compensated but is
‘regional’ compensated by the strength of the
lithosphere. Also, note the downwarps to either
side of the big island caused by the load that downflexes the lithosphere (see Fig. 9.10).
Oceanic island volcanic loads
compared to seismic evidence
Modelling gravity over Hawaiian Chain:
finding best elastic plate thickness of 30 km
4 km deep trenches on Venus, is this
subduction ?
What is a buoyancy force ?
A buoyancy force arises when a solid block (boat, mountain) is placed into an (ideal)-liquid.
The buoyancy force is specified by Archimede’s Principle which states: the decrease in weight
of the block equals the weight of the liquid displaced by the submerged portion of the body.
An ideal-liquid is simple in that the only force it transmits is pressure which means that an
ideal-liquid CANNOT support any shear stresses. The pressure in a liquid is: P= depth * liquiddensity * little-g (Passcals or N/m2 ).
The sign of the density contrast between the block and the liquid determines whether the
block floats (block density less than liquid) or sinks (block density greater than liquid).
Note below that the weight of the block (a) decreases as the block is lower into the fluid (b)
and when the block is floating (c) the effective weight becomes zero as the buoyancy force
supports the weight of the block!.
Examples of bouyancy
An object can change it density to
change buoyancy force and move
up or down in gravity field.
Layered floating blocks and gravity
Figure 9.3 shows two blocks with the same
density that are floating in a liquid. Note
that when a few layers of block (a) are
removed, hence reducing the blocks height
(mass), then the block floats at a new level
to maintain isostatic equilibrium. Also, note
that block (b) extends less above the liquid
surface because LESS fluid volume
displaced by the small weight of the shorter
block. One could also view this plot as
adding layers (mass) to block (b) which
causes block (a) to float with a higher
surface and deeper root.
Measuring gravity at points A,B,C finds no
change in gravity! This is because the
blocks are all in isostatic equilibrium which
means that the total mass of the block and
liquid beneath each blocks is the same! (I
am missing a detail here, what is that?)
Remember that for the blocks to float (not
sink), the liquid must be denser than the
blocks. Note the higher block’s roots
extends deeper into the fluid.
Deriving the isostatic mass and height equations
To calculate the isostatic balance between two blocks (A and B), the height and
weight (force) equations are derived. To do this, first the upper-level and
lower-level must be defined. We assume that the atmosphere is always at the
top of each block and that its weight, hence density, is negligible (assume
zero). One must always choose the upper-level to be ABOVE the top of either
block! For the lower-level, one must always choose this depth to be at the
base of the deepest block. We also assume that the asthenospheric ‘fluid’ that
the blocks float in is at the base of each block.
Specifying the height and weight equations
The Weight and Height equations are defined as:
Weight : g (  air hair  1h1   2 h2  3 h3   asth hasth )blk . A  g (  air hair  1h1   2 h2   asth hasth )blk . B
which simplifes to : ( 1h1   2 h2  3 h3   asth hasth )blk . A  ( 1h1   2 h2   asth hasth )blk . B
Height : (hair  h1  h2  h3  hasth )blk . A  (hair  h1  h2  hasth )blk . B
We now have two equations and many unknowns that cannot be solved algebraically. But,
if one knows densities and block layer thickness, we have 2 equations and 1 unknown and
can uniquely solve the algebraic system of equations.
Example: add 2 km ice to top of block
Block A
Block B
To the right block B,
two km of ice are
added. This added mass
causes the bottom of
the block to sink into
asthenosphere by ha
(km) as shown by the
right side block.
(0* hair  2*3  2.7 *30  3.1*70  3.2* hasth )blk . A  (0.9* 2  2*3  2.7 *30  3.1*70)blk .B
Note this equation can be greatly simplied by removing common terms
3.2* hasth  0.9* 2
hasth 
0.9* 2
 0.56 km
3.2
Height Eqn : hasth  hair  2 km
The addition of 2 km of ice causes block B to sink 0.56 km. Thus, using the height
equation, the former air/sediment surface sank by 2 - 0.56 = 1.44 km.
Isostasy: filling a lake with sediment
Notice there are two
unknowns in this problem:
the thickness of the new
sediments (hs ) and the
amount the block sinks into
the asthenosphere (ha ).
Amazing! Due to isostasy
the 2 km deep lake actually
filled with 3.14 km of
sediment.
Weight : 1.0 * 2  3.2ha  1.8 * hs
3.2 * ha  1.8 * hs  2
3.2 * ha  1.8 * (2  ha )  2
Height : 2  ha  hs
ha  1.14 km
hs  2  1.14  3.14 km
One equation and
two unknowns!
Back substitute hs
Height equation
Airy and Pratt end-member isostasy models
This was a big debate in 1855 after a British gravity/geodetic survey in India.
Mr. Pratt suggested that mountains do NOT have roots but instead the topography is
compensated by a less dense (hence lighter) block. And, that all topography was
compensated at the same depth.
Mr. Airy suggested that mountains had thick low density roots supporting mountains and
that the depth of compensation was NOT constant, but the density of the blocks was
constant.
TIME TO TEST THE HYPOTHESES!
Who was correct ?
Airy was mostly correct about what supports large (wide)
mountains, but it took until the 1970’s to prove this with
seismic work that measured the thickness of the crust and
lithosphere beneath mountains.
Pratt was correct in that the difference between the low
standing ocean basins and the high standing continents is
partially due to the fact that oceans have dense gabbroic
composition crust whereas continents have lighter less
dense ‘Andesitic’ composition crust.
Test: what are density variations between blocks
Free-air/Bouguer/Airy-isostatic gravity with and without compensation
Extra mass problem Free-air
Bouguer
Airy isostatic
Airy comp.
(0) w/ edge effect
(-)
0
No comp. (regional)
(+)
(0)
(+) with edge effect
Important is to know WHY the gravity effects of the two mass anomalies makes the
above table true. What happens if I remove the area (mass) above the red line from the
blocks ?
Viscous mantle (asthenosphere) response to loading
UM viscosity
Glacial isostatic adjustment
Gravity modelling: mid-ocean ridge and passive margin
Gravity over Amazon delta
Continental yield strength and earthquake depths
The strength (flow stress) of the lithosphere
is given by Byerlee’s brittle failure law and
measured flow law for different indicated
minerals for assumed geotherm for 100 Ma
lithosphere.
Earthquake in continents occur mostly at
<25 km depth, but <50 km in oceans.
Difference in earthquake depths
manifests variation in strength with depth
in oceanic and continental
crust/lithosphere
Changing strength of oceanic lithosphere in time
(a) Note that as the lithosphere gets older (right to left ‘flag plots’), it gets
stronger due to thermal cooling of lithosphere.
(b) Integrated strength of lithosphere (gray area under (a) curves) increase in
time.
Water height in oceans is proportional to gravity field
By radar mapping the ocean
surface, the gravity field is
measured (also current
pressures).
This is because the liquid
ocean surface is
perpendicular to the gravity
field.
Figure of the earth
Terms to know:
•Spherical radius
•Equatorial radius
•Polar radius
•Reference ellipsoid
•Geoid (Gravitational Potential)
Non-spherical shape and centrifugal
force makes the gravity vary as IGF
equation.
Also, the earth rotates once a day
around its spin axis to make forces.
What force makes the Earth a
flattened ellipsoid with a smaller
polar radius ?
Measured figure of earth
Isostasy quotes (1)
That part of the surface of any heavy body will become more distant from the centre of its
gravity which becomes of greater lightness. The earth therefore, the element by which the
rivers carry away the slopes of mountains and bear them to the sea, is the place from which
gravity is removed: it will make itself lighter……The summits of the mountains in course of
time rise continually (da Vinci, 1505).
The mountains, I think, are to be explained chiefly as due to thermal expansion of material
at depth, whereby the rock layers near the surface are lifted up. The uplifting does not
mean the inflow or addition of material at depth, the void within the mountain
compensates for the overlying mass (Boscovich, 1755).
The state of the Earth’s crust lying upon the lava may be compared with perfect correctness
to the state of a raft of timber floating upon water; in which, if we remark one log whose
surface floats much higher than the upper surfaces of the others, we are certain that its
lower surface lies deeper in the water than the lower surfaces of the others (Airy 1855).
The amount of matter in any vertical column drawn from the surface to a level surface
below the crust is now and ever has been, approximately the same in every part of the
Earth (Pratt, 1861).
Isostasy quotes (2)
The crust must be in a condition of approximate hydrostatical equilibrium, such that
any considerable addition of load will cause any region to sink, or any considerable
amount deduced off an area will cause it to rise……..the crust is analogous to the case of a
broken-up area of ice, refrozen and floating upon water. (Fisher, 1881).
The hypothesis (interior contraction by secular cooling) is nothing but a delusion and a snare,
and the quicker it is thrown aside and abandoned, the better it will be for geological science
(Dutton, 1882).
In an unpublished paper I have used the terms isostatic and isostasy to express that
condition of the terrestrial surface which follow from the floatation of the crust upon a liquid
or highly plastic substratum…….isobaric would have been a preferable term, but it is
preoccupied in hypsometry………….For this condition of equilibrium of figure, to which
gravitation tends to reduce a planetary body, irrespective of whether it is homogeneous or
not, I propose the name isostasy (Dutton 1882).
An Airy theory is untenable……The Pratt theory is the only one so far that is sound……..It
seems safe to assert that the teory of isostasy has been proven……….. (Bowie 1927).
Convenience of computation, and perhaps tradition, rather than any physical probability, has
been the chief reasons for the attention given to Pratt’s hypothesis, instead of Airy’s (Jeffreys
1926).
Isostasy quotes (3)
Various geological observations and deductions, which the geologists regard as established
facts (horizontal shortening), seems inconsistent with the isostatic theory (Chamberlin 1932).
Geologists often ask too much of the principle of isostasy. When they find that it will not
explain all earth movements, they think it is not a true principle (Reid 1922).
Mountains, mountain ranges, and valleys of magnitude equivalent to mountains, exist
generally in view of the rigidity of the Earth’s crust; continents and plateaus and oceanic basins
exist in virtue of isostatic equilibrium in a crust heterogeneous wrt density (Gilbert 1889).
The success of the isostatic hypothesis in reducing gravity anomalies is to show that isostatic
adjustment in the earth’s crust is nearly perfect (Gilbert 1913).
The excesses and deficits of mass……..will be a more accurate measure of the capacity of the
rigid crust to carry without viscous yielding loads which have borne through geological time,
hidden loads whose magnitudes in many regions appear to mask by contrast the present relief
between mountains and valleys……….The deep zone is the hydraulic agent which converts the
gravity of the excess matter in the heavy column into a force acting upwards against the lighter
column….By this means even the continental interiors are kept in isostatic equilibrium with the
distant ocean basins (Barrell 1914).
No question that a mountain’s load is distributed beyond the area of its base (Putnam 1935).
Comparison of astronomic and geodetic determined latitude
The keynote of isostasy is a working towards equilibrium. Isostasy is not a process that
disturbs equilibrium, but one that restores equilibrium.
Gravity deficient and the Himalaya’s
Airy’s floating table-land hypothesis
(1855)
B block  b liquid
liquid
b

B
block
Global elastic thickness
Free air gravity
Global free air gravity from satellite
Crustal
thickness:
Airy, Pratt
or both or
NOT?
Crustal thickness
in western US
from Earthscope
array data
Free-air gravity through Hawaiian Islands